Applications · Atmospheric science
Chaos in weather and climate
Edward Lorenz's 1963 paper truncated atmospheric convection to three ODEs and discovered that microscopic differences in initial conditions explode into completely different long-term states. The 'butterfly effect' became the canonical example of chaos and set the practical predictability horizon for weather at roughly two weeks.
x-coordinate of two Lorenz trajectories starting ε apart
log₁₀ |Δx(t)| vs t — slope = leading Lyapunov exponent
Sensitive dependence on initial conditions
Two Lorenz trajectories start ε apart. For small ε the divergence grows exponentially: |Δx(t)| ≈ ε · e^{λ₁ t}. Once the difference saturates at the attractor diameter (~30), nonlinearity takes over and the two trajectories become uncorrelated. This is the mathematical content of the 'butterfly effect'.
Lorenz '63 and the predictability horizon
λ₁ ≈ 0.906 day⁻¹ in the original Lorenz model. A small initial error doubles in size every 0.77 days. Realistic atmospheric models extend the horizon to around 14 days, beyond which any forecast is statistical at best.
Ensemble forecasting
Modern weather forecasts run dozens of integrations from slightly perturbed initial conditions. The spread of the ensemble at a given forecast lead time is a direct chaos measurement: a tight ensemble means a predictable atmospheric state; a wide one means SDIC has bitten.
Lorenz '96 and '63
Beyond the famous 3-variable system, Lorenz constructed an N-variable model in 1996 that captures spatially extended chaos and is widely used as a testbed for data-assimilation and machine-learning weather emulators.
Climate vs weather
Climate (long-term statistics of the attractor) can be stable even when individual weather trajectories are chaotic; this is the central insight that lets climate models be useful despite the underlying chaos.
See also
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Quick quiz
Test yourself on weather
8 multiple-choice questions. Pick an answer for each, then submit to see explanations.
Q1.Edward Lorenz's '63 paper introduced:
Q2.The 'butterfly effect' refers to:
Q3.Practical weather-forecasting horizon in current models is about:
Q4.Ensemble forecasting works by:
Q5.Lorenz '96 model is:
Q6.Climate is to weather as:
Q7.Lorenz's classical parameters σ, ρ, β are:
Q8.Why doesn't chaos prevent climate forecasting entirely?